I understand it was a matter of simply, openly paying $300 to get out of the draft, but did doing so have any sort of appreciable social impact, after the war, on the people who did?
Just wrote a paper relating to this!
There was both a social stigma and even some physical danger in buying your way out of the draft (either through commutation or by hiring a substitute). For example, there was a popular tune of the time called "We Are Coming, Father Abraham" (a parody of an earlier recruiting song of the same name) which included lines like:
We are coming, Father Abraham,
Three hundred dollars more
We'll stay at home and take our ease,
For fighting's such bore;
We can't all go and meet the foe,
For who'll attend the ladies?
The Exempt Brigade will stay at home
To nurse and dress the babies!
Going even further, during the New York draft riots in July 1863, rioters were a little angry at anyone that they thought might have bought their way out of the draft:
Rioters sacked the homes of several prominent Republicans and abolitionists. With shouts of "Down with the rich" and "There goes a $300 man" they attacked well-dressed men who were incautious enough to show themselves on the streets.
During the election of 1884 Grover Cleveland was criticized for purchasing a substitute during the Civil War, a Polish immigrant George Benninsky who he paid $150. He did so because he was the sole financial support of his mother and younger sisters. The Civil War being won less than nineteen years prior to the 1884 election was very much in public memory, being a "$300 Man" and the nominee of the Democratic Party, which made up a vast part of the Confederate rebellion certainly hurt his image(along with the rumors of an out of wedlock daughter). However his reputation for honesty, especially compared with the corrupt Sen.James Blaine lead to his victory. Veterans of both sides of the war prided themselves with their service, they formed powerful veterans groups like the GAR , officers were often addressed publicly by their military titles decades after the conflict. Those who had bought their way out found themselves in a special isolated corner of their generation, though not often publicly damned they were looked down upon. Theodore Roosevelt who admired his father found it humiliating that he had, in TR's perception shirked his duty as an American citizen and as a man.